Question

what is the area of triangle qrs? 10 square units 7 square units 9 square units 13 square units

Explanation:

Step1: Identify base and height

Let's consider the base along the vertical - like line and height along the horizontal - like line. If we consider the line segment between the points with the same x - coordinate as the base. Counting the grid units, the base of triangle QRS is 5 units (vertical distance from the highest to the lowest y - value among the vertices) and the height is 2 units (horizontal distance between the vertex with the most left - hand x - value and the vertex with the most right - hand x - value among the vertices).

Step2: Apply area formula

The area formula for a triangle is $A=\frac{1}{2}\times b\times h$. Substituting $b = 5$ and $h=2$ into the formula, we get $A=\frac{1}{2}\times5\times2$.

Step3: Calculate the area

$A=\frac{1}{2}\times5\times2 = 5$ square units. But if we use another way. Let's enclose the triangle in a rectangle. The rectangle that encloses triangle QRS has length 5 units and width 4 units. The area of the rectangle is $A_{rect}=5\times4 = 20$ square units. Then, we subtract the areas of the three right - angled triangles that are formed within the rectangle but outside of triangle QRS.
The three right - angled triangles:

1. One triangle with base 2 and height 2, its area $A_1=\frac{1}{2}\times2\times2 = 2$ square units.
2. Another triangle with base 3 and height 2, its area $A_2=\frac{1}{2}\times3\times2=3$ square units.
3. The third triangle with base 2 and height 5, its area $A_3=\frac{1}{2}\times2\times5 = 5$ square units.

The sum of the areas of these three triangles is $A_{sum}=2 + 3+5=10$ square units.
The area of the rectangle is 20 square units. So the area of triangle QRS is $A = 20-10=10$ square units.

Answer:

10 square units